Towards manipulating relativistic laser pulses with micro-tube plasma lenses

Efficient coupling of intense laser pulses to solid-density matter is critical to many applications including ion acceleration for cancer therapy. At relativistic intensities, the focus has been mainly on investigating various laser beams irradiating initially overdense flat interfaces with little or no control over the interaction. Here, we propose a novel approach that leverages recent advancements in 3D direct laser writing (DLW) of materials and high contrast lasers to manipulate the laser-matter interactions on the micro-scales. We demonstrate, via simulations, that usable intensities ≥1023 Wcm−2 could be achieved with current tabletop lasers coupled to micro-engineered plasma lenses. We show that these plasma optical elements act as a lens to focus laser light. These results open new paths to engineering light-matter interactions at ultra-relativistic intensities.

Despite recent progress, controlling light-matter interactions at relativistic intensities has not been extensively explored for two main reasons. First, the manufacturing of advanced micro-and nano-structures has been the domain of other scientific disciplines and their use is confined to intensities below the damage threshold. Second, high-intensity short-pulses are inherently preceded by nanosecond scale pedestals 45 that can destroy or substantially modify any guiding structure due to hydrodynamic expansion. However, with recent advancement in laser technology and laser pulse cleaning techniques it is now possible to substantially reduce the undesired pedestals. Laser to pedestal contrast ratios higher than 10 10 have been achieved using cross-polarized wave generation (XPW) technique 46 . Higher contrast can also be achieved by using the plasma mirror method [47][48][49] . Furthermore, highly repeatable structures with features as small as 100 nm can be easily manufactured using 3D direct laser-writing (DLW) 50 . The small size of the light guiding features and the short duration of the current generation of high contrast laser (tens of femtoseconds) make it possible to optimize the interaction using realistic and computationally manageable particle-in-cell (PIC) simulations.
We show via PIC simulations that, one of the most critical parameters-the laser intensity, can be manipulated using hollow micro-cylinders. The structured target exhibits optical resonance. When a high contrast intense laser pulse is input into the Micro-Tube-Plasma (MTP) lens, it is significantly intensified. The new technique is able to boost light intensity up to 10 23 Wcm −2 based on today's high power laser systems, allowing for the exploration in the exotic near-QED (quantum electro-dynamics, I > 10 22 W/cm 2 ) regime. Further, by manipulating the laser intensity, controlling or maximizing the outcome of laser-plasma interaction becomes possible. We show Scientific RepoRts | 6:23256 | DOI: 10.1038/srep23256 that the interaction of an intense laser beam with flat target via an MTP lens can increase the on-target laser intensity by concentrating the energy in a smaller focal spot. As a result, the generation of secondary electron, proton, and γ -ray beams are enhanced.

Results
Light intensification in a micro-tube plasma target. The proposed scheme is shown in Fig. 1a. The target consists of periodic hollow micro-cylinders attached to a flat substrate. These guides can now be printed using two-photon lithography process available on most commercial 3D printers with a featured resolution around 200 nm. With many identical micro-tubes, the laser will enter one of them with a good and certain opportunity.
To get insight into the interaction of intense light with micro-tubes, we have carried out 3D PIC simulations of a high-contrast laser pulse with a single tube (Fig. 1b). A laser beam with a duration of 40 fs and intensity of 5.3 × 10 21 Wcm −2 enters the target via a micro-tube of 4.8 μm diameter. Such laser pulses are available at current laser facilities 51 . Comparing the beam intensities outside and inside the tube reveals substantial enhancement. The laser intensity is boosted by a factor of ~3 inside the tube. A peak intensity of 1.5 × 10 22 Wcm −2 is reached after the pulse has propagated a distance of 8 μm from the entrance aperture, which is much smaller than the pulse Rayleigh length (~30 λ 0 in our case). Increasing laser beam intensity requires increasing the total laser energy, shortening the pulse duration, and/or decreasing the laser focal spot. Cones have been found to be able to enhance incident laser, but no physical explanation is presented 52 . Our detailed analyses of the spatial and temporal characteristics of the laser indicate that the light intensification is due to tight focusing of the beam. The left and right plots in Fig. 1c show the laser spatial distributions in the transverse y − z plane averaged along the propagation axis x. The dashed white line indicates the tube boundary. It is evident from these spatial profiles that the micro-tube acts as a focusing optical element. A tight focal spot with a full-width-half-maximum (FWHM) of 1.8 μm is achieved inside the tube compared to 2.9 μm (FWHM) for the input beam. Temporal snapshots of the beam did not reveal any pulse duration shortening (Fig. 1c). Both input and in-tube laser pulses show similar oscillatory motions in time.

Mechanism and scaling of light intensification.
We performed a series of 3D PIC simulations to systematically study light intensification in a hollow micro-cylinder with a fixed diameter (4.8 μm). The laser intensity was varied from 10 17 Wcm −2 to 10 23 Wcm −2 . The peak laser intensity inside the tube at each simulation time is recorded for all input pulses. The light intensification factor is determined as the ratio of the in-tube to the input pulse intensity (η = I in-tube /I input ). This quantity is plotted in Fig. 2a as a function of the input laser intensity. Three distinct regimes of intensification are clearly observed.
Diffraction regime. In this regime, the light intensification is independent on the laser intensity. As the electron density distribution in Fig. 2b indicates, there is no background plasma inside the tube. The interaction of the laser pulse with the tube is predominantly determined by the size of the aperture. As a result the narrowing of the focal spot is mainly due to diffraction. The intensification happens in the near-field Fresnel region, since the aperture size is large compared to the light wavelength. The intensification factor is constant at around η ≈ 2.6 for input laser amplitudes below I input ∼ 10 19 Wcm −2 . In laser-wakefield acceleration, capillary tubes filled with underdense plasma have been used for pulse guiding, though no light intensification was observed.
Depletion regime. In the second regime, the laser field is strong enough to pull-off a considerable portion of electrons from the wall of the micro-tube. These laser-induced electrons form a low density cloud, below one tenth of the critical density (Fig. 2c). These electrons are distributed in an unorganized fashion (see also in Fig. 2e). In addition to being diffracted, the laser field is depleted by the under-dense plasma electrons, leading to a slightly lower intensification factors compared to the ones in the diffraction regime. Focusing regime. For laser intensities higher than I ≈ 6 × 10 21 Wcm −2 , we observe a monotonic increase of the in-tube pulse intensity. In this domain of LPI, the laser amplitude is high enough to pull-off a substantial fraction of the electrons from the walls of the micro-tube. These electrons stay together, forming periodic dense bunches, as shown in Fig. 2d,f. The averaged electron density in the tube is shown in Fig. 2e. At the inner boundary, in the vicinity of the wall, the formed plasma is over-dense (i.e. with density equal or higher to the critical density). Within this laser-induced over-dense layer, the outer edge of the laser field decays rapidly to zero in a skin depth. This layer changes the tube geometry by acting as a new tube boundary. The laser pulse sees a diameter-reduced tube in this case and the laser energy is confined to smaller focal spot.
The plasma density is lower on axis and higher when approaching the boundary. The V-shaped plasma density profile also contributes to the focusing for the propagation laser pulse. The intensification factor increases with input intensity. A laser beam with an initial intensity of 2.4 × 10 22 Wcm −2 could be boosted to a peak intensity 10 23 Wcm −2 using the MTP device. The experimentally unexplored near-QED (quantum electro-dynamics) regime could be within reach on the existing laser facilities using micro-tube plasma devices coupled to primary targets.
Thresholds between three regimes. The transition from the diffraction to the depletion regime occurs when the laser pulse starts to create plasma in the tube. As electrons are dragged out of the tube by the transverse laser field, they are also accelerated forward directly by the Lorentz force, specifically by direct laser acceleration (DLA) mechanism. Efficient DLA requires that an electron is accelerated to a relativistic velocity in half laser period. Considering an electron being accelerated by DLA, if the laser is non-relativistic, the electron longitudinal velocity would be much smaller than the speed of light. It will slip to the deceleration phase in less than half of the laser period, where the direction of the laser field is opposite to the one in the accelerating phase. The electron is driven outwards and scattered off immediately. In this case, electrons will not be able to reach the central area of the tube and there is no background plasma in the tube. Hence it is when the laser field at the tube boundary becomes relativistic that the electrons can be detached and stay in the tube. This gives Here a = eE L /m e ω 0 c is the dimensionless laser electric amplitude and = ∆ + ∆ r y z 2 2 is the vertical distance from the propagation axis (the inner tube boundary is located at r 0 ). For the pulse profile and tube size we employed, Eq. 1 produces a 0 ≥ e(e ≈ 2.72, the Euler number), corresponding to a peak intensity of I 0 ≈ 1.6 × 10 19 Wcm −2 , in good agreement with Fig. 2a.
The threshold between the depleting and focusing regimes corresponds to when the electron density near the inner wall becomes over-dense, i.e., . For the laser profile and tube radius we use in simulations, the dimensionless laser amplitude can be written as π = a a e / r 0 0 . The electron density at the boundary is obtained by balancing the two fields. This gives, The laser amplitude required to create critical plasma density is a thr ≈ 53(I 0 ≈ 6 × 10 21 Wcm −2 ) for r 0 = 3λ 0 , which is consistent with the laser amplitude observed in Fig. 2a.

Applications of the MTP target.
The fact that the laser beam maintains its space-time integrity and is enhanced in intensity suggest that 3D printed structures could be used, as an intermediary micro-optical element, to increase the intensity on a flat foil at the end of the tunnel and thus manipulate or maximize the outcome of the LPI. As a case study, we carried out an investigation using an input laser intensity of 5 × 10 22 Wcm −2 (a 0 = 150), which lies in the focusing regime. We compare two cases: 1) laser beam interacting with a traditional carbon-hydrogen (CH) flat target (Fig. 3a), 2) laser beam interacting with an identical flat target coupled to a 8 μm long 3D printed micro-tube (Fig. 3b). The introduction of the intermediary micro-optical plasma elements increases the peak intensity on the secondary flat interface to 4.3 × 10 23 Wcm −2 . We observe significant enhancements in terms of maximum and total energy of laser-induced electron, proton, and γ-ray beams when the micro-tube plasma lens is used (Fig. 3c-e). Due to MTP lens, the maximum electron beam energy is increased from ~200 MeV to ~400 MeV. The laser-to-electron energy conversion efficiency for high-energy electrons (> 50 MeV) is enhanced by a factor of 8 in the compound target. Consequently, the γ-ray beam production is enhanced by a similar factor as the high-energy electrons are decelerated by the radiation back-reaction. With a micro-tube plasma lens, the maximum photon energy reaches 50 MeV compared to only 20 MeV with a traditional flat target. The foil at the end of the micro-tube is strongly deformed due to the increased laser pressure of the intensified laser pulse. The traditional target is only slightly bent, indicating a much smaller hole-boring velocity due to the weaker on-target laser intensity. The maximum proton energy obtained with compound targets is ~350 MeV compared to only 100 MeV for flat targets for the same input laser intensity. The total number of energetic protons increases fivefold with light intensification.

Discussion
The last 30 years has seen the development of ever increasing peak intensity from pulsed lasers. The quest for extremely intense laser pulses is largely motivated by the possibility of reaching experimentally unexplored regimes of light-matter interactions as well as the production of intense ion beams for cancer therapy. To increase the light intensity, one or a combination of the pulse parameters (energy, duration, and spot size) must be manipulated accordingly. Increasing the energy and/or shortening the pulse require using large optical elements due to damage threshold. Other nonconventional approaches to altering the incident laser pulse such as self-focusing 53 , pulse modification with capillaries 54,55 , coherent focusing of harmonics 56 , and flying mirror 57 have been proposed. Additionally, structured interfaces such as nano-cells and microcylinder arrays have shown an advanced level of control over the secondary processes in the LPI 58 . Here, we have outlined a novel approach that leverages recent advances in micro-engineering of materials and high contrast lasers to boost light intensity using hollow micro-cylinders. The use of the micro-optical elements provides another degree of freedom that makes it possible to micro-engineer laser plasma interactions. By controlling and adjusting the spatial dimensions (aperture size, length, thickness,..) of the MTP targets, the pulse intensity can be manipulated. Hence various LPI applications can be tuned for optimal performance. Further studies show that they also act as an electromagnetic guide for secondary particle beams (will be presented elsewhere). These new results will open new paths towards micro-engineering laser plasma interactions that will benefit high field science, laser-based proton therapy, laser and particle beams, near-QED physics, nuclear physics, and relativistic nonlinear optics.

Methods
Numerical modeling of the laser interaction with micro-tube plasma target. We simulate laserplasma interaction with the full-3D particle-in-cell code VLPL 59 . In a simulation box of 40λ 0 × 12λ 0 × 12λ 0 (λ 0 = 0.8 μm is the laser wavelength) in x × y × z directions, a laser pulse polarized in the y direction enters from the left boundary along the x direction. The laser field amplitude has a profile of ω = where ω 0 is the laser frequency, a = eE L /meω 0 c is the dimensionless laser electric amplitude. Here e, m e are fundamental charge and electron mass, E L is the laser electric field, c is the speed of light in vacuum, respectively. The pulse duration and spot size are defined as τ 0 and σ 0 . A single carbon tube is placed 10 λ 0 from the left boundary, with a diameter of 6λ 0 . The electron density of the tube when fully ionized is n e = 300 n c and the thickness of the wall is λ 0 . The whole target is cold and pre-ionized. To see the full evolvement of a laser pulse in the tube, we set the tube length to be sufficiently long. The cell size is 0.02λ 0 × 0.1λ 0 × 0.1λ 0 to resolve the fine structure and the time step is Δt = 0.008 T 0 to suppress the numerical instability for high plasma density When simulating at intensities as high as 10 23 W/cm 2 , the recoil force on a radiating electron, i.e., the radiation reaction (RR) force must be included in LPI [59][60][61][62][63] . In that case, the code VLPL employs a quantum-electrodynamic (QED) model to calculate the emitted photons and the RR force [64][65][66] . In our simulations, we turned on the QED model when a 0 ≥ 100.